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Physica Scripta

From quantum carpets to quantum suprematism—the probability representation of qudit states and hidden correlations

Margarita A. Man’ko, Vladimir I. Man’ko

Published Jun 26, 2018• 23 citations
Publisher

Abstract

Abstract We develop the tomographic-probability picture where quantum states of qudits (spin- j systems, N -level atoms) are described by means of sets of fair classical probability distributions associated with properties of sets of dichotomic classical random variables. We formulate the superposition principle of pure quantum states as nonlinear addition rule of standard probabilities. The invertible map of the density matrices of qudit states onto the probability distributions is illustrated by their geometric representation in terms of triangles and squares called Malevich’s squares. We discuss an analogy of the squares with the quantum carpet picture. New entropic inequalities obtained due to the introduced map of the density matrices onto Malevich’s squares are studied. The correlations in systems without subsystems (hidden correlations) are discussed.

DOI
10.1088/1402-4896/aacf24

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Abstract We develop the tomographic-probability picture where quantum states of qudits (spin- j systems, N -level atoms) are described by means of sets of fair classical probability distributions associated with properties of sets of dichotomic classical random variables. We formulate the superposition principle of pure quantum states as nonlinear addition rule of standard probabilities. The invertible map of the density matrices of qudit states onto the probability distributions is illustrated by their geometric representation in terms of triangles and squares called Malevich’s squares. We discuss an analogy of the squares with the quantum carpet picture. New entropic inequalities obtained due to the introduced map of the density matrices onto Malevich’s squares are studied. The correlations in systems without subsystems (hidden correlations) are discussed.
We study an analog of Bayes’ formula and the nonnegativity property of mutual information for systems with one random variable. For single-qudit states, we present new entropic inequalities in the form of the subadditivity and condition corresponding to hidden correlations in quantum systems. We present qubit states in the quantum suprematism picture, where these states are identified with three probability distributions, describing the states of three classical coins, and illustrate the states by Triada of Malevich’s squares with areas satisfying the quantum constraints. We consider arbitrary quantum states belonging to N-dimensional Hilbert space as ( N 2 − 1 ) fair probability distributions describing the states of ( N 2 − 1 ) classical coins. We illustrate the geometrical properties of the qudit states by a set of Triadas of Malevich’s squares. We obtain new entropic inequalities for matrix elements of an arbitrary density N×N-matrix of qudit systems using the constructed maps of the density matrix on a set of the probability distributions. In addition, to construct the bijective map of the qudit state onto the set of probabilities describing the positions of classical coins, we show that there exists a bijective map of any quantum observable onto the set of dihotomic classical random variables with statistics determined by the above classical probabilities. Finally, we discuss the physical meaning and possibility to check derived inequalities in the experiments with superconducting circuits based on Josephson junction devices.
Review of tomographic probability representation of quantum states is presented both for oscillator systems with continious variables and spin–systems with discrete variables. New entropy–information inequalities are obtained for Franck–Condon factors. Density matrices of qudit states are expressed in terms of probabilities of artificial qubits as well as the quantum suprematism approach to geometry of these states using the triadas of Malevich squares is developed. Examples of qubits, qutrits and ququarts are considered.
This paper proposes an alternative geometric representation of single qudit states based on probability simplexes to describe the quantum properties of noncomposite systems. In contrast to the known high dimension pictures, we present the planar picture of quantum states, using the elementary geometry. The approach is based on, so called, Malevich square representation of the single qubit state. It is shown that the quantum statistics of the single qudit with some spin j and observables are formally equivalent to statistics of the classical system with N 2 − 1 random vector variables and N 2 − 1 classical probability distributions, obeying special constrains, found in this study. We present a universal inequality, that describes the single qudits state quantumness. The inequality provides a possibility to experimentally check up entanglement of the system in terms of the classical probabilities. The simulation study for the single qutrit and ququad systems, using the Metropolis Monte-Carlo method, is obtained. The geometrical representation of the single qudit states, presented in the paper, is useful in providing a visualization of quantum states and illustrating their difference from the classical ones.
We consider the probability distributions, spin (qudit)-state tomograms and density matrices of quantum states, and their information characteristics, such as Shannon and von Neumann entropies and q-entropies, from the viewpoints of both well-known purely mathematical features of nonnegative numbers and nonnegative matrices and their physical characteristics, such as entanglement and other quantum correlation phenomena. We review entropic inequalities such as the Araki–Lieb inequality and the subadditivity and strong subadditivity conditions known for bipartite and tripartite systems, and recently obtained for single qudit states. We present explicit matrix forms of the known and some new entropic inequalities associated with quantum states of composite and noncomposite systems. We discuss the tomographic probability distributions of qudit states and demonstrate the inequalities for tomographic entropies of the qudit states. In addition, we mention a possibility to use the discussed information properties of single qudit states in quantum technologies based on multilevel atoms and quantum circuits produced of Josephson junctions.